Exercise 4, § 3

Suppose that K is a field of characteristic $p > 0$.

(a) Let H be a basis of the Lie subalgebra L(X) of A(X). Show (using the isomorphism $U(L(X)) \to A(X)$ and the Poincaré–Birkhoff–Witt Theorem) that the elements $h^{p^n}$ ($h \in H, n \in \mathbf{N}$) are linearly independent over K.

(b) If m is an integer $\geqslant 0$, let $L_m$ denote the submodule of $A(X)$ with basis the $h^{p^n}$, where $h \in H, n \leqslant m$. Show that $L_m$ is a Lie subalgebra of $A(X)$ and that, if $a \in L_{m-1}$, then $a^p \in L_m$ (argue by induction on m and use the Jacobson formulae, cf. Chapter I, § 1, Exercise 19). Deduce that $L_m$ does not depend on the choice of H and that it is the Lie subalgebra of $A(X)$ generated by the $x^{p^n}$, where $x \in X, n \leqslant m$.

(c) Let $L(X, p)$ be the union of the $L_m$ for $m \geqslant 0$. Show that $L(X, p)$ is the smallest Lie $p$-subalgebra of $A(X)$ containing X (cf. Chapter I, § 1, Exercise 20); it admits as basis the family of $h^{p^n}$, where $h \in H, n \in \mathbf{N}$.

(d) Let $\tilde{U}(L(X, p))$ be the restricted enveloping algebra of $L(X, p)$, cf. Chapter I, § 2, Exercise 6. Show that the injection of $L(X, p)$ into $A(X)$ can be extended to an isomorphism of $\tilde{U}(L(X, p))$ onto $A(X)$. (Use the Poincaré–Birkhoff–Witt Theorem and the exercise quoted above.) Deduce that $L(X, p)$ is the set of primitive elements of $A(X)$, cf. § 1, Exercise 12.

(e) Let $f$ be a mapping of X into a Lie $p$-algebra g. Show that $f$ can be extended uniquely to a $p$-homomorphism $F : L(X, p) \to g$. (Begin by extending $f$ to an algebra homomorphism of $A(X)$ into $\tilde{U}g$.)

(The Lie $p$-algebra $L(X, p)$ is called the free Lie $p$-algebra on the set X.)